Operations on one matrix
Simple row operations
Switching two rows
You can switch any two rows in a matrix without changing the value of the matrix.
Multiplying a row by a constant
You can multiply any row in a matrix by any non-zero constant without changing the value of the matrix. We often call this value a scalar because it “scales” the values in the row.
Adding a row to another row
It’s also acceptable to add one row to another. Be careful that this doesn’t consolidate two rows into one. Instead, we replace a row with the sum of itself and another row.
Pivot entries and row-echelon forms
Now that we know how to use row operations to manipulate matrices, we can use them to simplify a matrix in order to solve the system of linear equations the matrix represents.
Pivot entries
A pivot entry(or leading entry, or pivot) is the first non-zero entry in each row. Any column that houses a pivot is called pivot column. Look at the picture below. All of the pivots are circled in blue.
Row-echelon form
A matrix is in row-echelon form (ref) if
- All the pivot entries are equal to 1.
- Any rows that consist of only 0s are at the bottom of the matrix.
- The pivot in each row sits in a column to the right of the column that houses the pivot in the row above it. In other words, the pivot entries sit in a staircase pattern, where they stair-step down from the upper left corner to the lower right corner of the matrix.
If a matrix is in ref form, and if, in each pivot column, the pivot entry is the only non-zero entry, then the matrix is in reduced row-echelon form(rref). Below is an example of putting the matrix into rref form using row operations.
Gauss-Jordan elimination
Finally, we’re at a point where we can start to solve a system using a matrix. To solve a system, our goal will be to use the simple row operations we learned earlier to transform the matrix into row-echelon form, or better yet, reduced row-echelon form.
- Optional : Pull out any scalars from each row in the matrix.
- If the first entry in the first row is 0, swap it with another row that has a non-zero entry in its first column. Ohterwise, move to step 3.
- Multiply through the first row by a scalar to make the leading entry equal to 1.
- Add scaled multiples of the first row to every other row in the matrix until every entry in the first column, other than the leading 1 in the first row, is a 0.
- Go back step 2 and repeat the process until the matrix is in reduced row-echelon form
Below is an example of solving the system using Gauss-Jordan elimination.
Numbers of solutions to the linear system
We already know how to solve linear systems using Gaussian elimination to put a matrix into reduced row-echelon form. But up to now, we’ve only looked at systems with exactly one solution. In fact, systems can have: one solution, or no solutions, or infinitely many solutions.
The unique solution
Infinitely many or no solutions
Pivot entries and free entries
You already know that the 1’s in the first and second row of matrix are called the pivot entries in the matrix. Any other non-zero values on the left side of the matrix, in this case the −3 in the second row, are called free entries.
